Inner product in $\wedge^{k}V^{n}$

Question

Given an inner product $\langle \cdot, \cdot \rangle$ in $V$, $n$-dimensional vector space, we may consider the following inner product in $\wedge^{k}V$ given by

$$\langle u_{1}\wedge...\wedge u_{k},v_{1}\wedge...\wedge v_{k} \rangle \colon= \det(\langle u_{i},v_{j}\rangle)_{k\times k}. $$

where $u_{1},..,u_{k},v_{1},...,v_{k} \in V$.

Which other inner product we can define in $\wedge^{k} V$ given an inner product in $V$?

Thanks in advance.